A080856 a(n) = 8*n^2 - 4*n + 1.

1, 5, 25, 61, 113, 181, 265, 365, 481, 613, 761, 925, 1105, 1301, 1513, 1741, 1985, 2245, 2521, 2813, 3121, 3445, 3785, 4141, 4513, 4901, 5305, 5725, 6161, 6613, 7081, 7565, 8065, 8581, 9113, 9661, 10225, 10805, 11401, 12013, 12641, 13285, 13945, 14621

Offset: 0

Author

Paul Barry, Feb 23 2003

Keywords

• nonn
• easy

Comments

The old definition of this sequence was "Generalized polygonal numbers".

Row T(4,n) of A080853.

{a(k): 0 <= k < 3} = divisors of 25. - Reinhard Zumkeller, Jun 17 2009

Let A be the Hessenberg matrix of order n, defined by: A[1,j]=1, A[i,i]:=4, (i>1), A[i,i-1]=-1, and A[i,j]=0 otherwise. Then, for n>=3, a(n-1)= coeff(charpoly(A,x),x^(n-2)). - Milan Janjic, Jan 27 2010

Also sequence found by reading the segment (1, 5) together with the line from 5, in the direction 5, 25,..., in the square spiral whose vertices are the generalized hexagonal numbers A000217. - Omar E. Pol, Nov 05 2012

For n > 0: A049061(a(n)) = 0, when the triangle of "signed Eulerian numbers" in A049061 is seen as flattened sequence. - Reinhard Zumkeller, Jan 31 2013

Links

• G. C. Greubel, Table of n, a(n) for n = 0..5000
• Milan Janjic, Hessenberg Matrices and Integer Sequences, J. Int. Seq. 13 (2010) # 10.7.8.
• Reinhard Zumkeller, Enumerations of Divisors.
• Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Crossrefs

Cf. A005408, A000124, A016813, A049061, A080853, A086514, A000125, A058331, A002522, A161701, A161702, A161703, A000127, A161704, A161706, A161707, A161708, A161710, A161711, A161712, A161713, A161715, A006261.

A060820 is another version (but the present sequence is the main entry).

A row of the array in A386478.

Programs

Maple

A080856:=n->8*n^2 - 4*n + 1: seq(A080856(n), n=0..100); # Wesley Ivan Hurt, Jul 16 2017

Mathematica

LinearRecurrence[{3, -3, 1}, {1, 5, 25}, 80] (* Vladimir Joseph Stephan Orlovsky, Feb 17 2012 *)

PARI

a(n)=8*n^2-4*n+1 \\ Charles R Greathouse IV, Sep 24 2015

Formula

G.f.: (1+2*x+13*x^2)/(1-x)^3.

a(n) = A060820(n), n>0. - R. J. Mathar, Sep 18 2008

a(n) = C(n,0) + 4*C(n,1) + 16*C(n,2). - Reinhard Zumkeller, Jun 17 2009

a(n) = 16*n+a(n-1)-12 with n>0, a(0)=1. - Vincenzo Librandi, Aug 08 2010

E.g.f.: (8*x^2 + 4*x + 1)*exp(x). - G. C. Greubel, Jun 16 2017

Extensions

Definition replaced with the closed form by Bruno Berselli, Jan 16 2013